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Vortex-induced vibrations of a rigid cylinder on elastic supports with end-stops, Part 1: Experimental results
Journal of Fluids and Structures 29 (2012) 62–78
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Vortex-induced vibrations of a rigid cylinder on elastic supports with end-stops, Part 1: Experimental results Sylvain Bourdier 1, John R. Chaplin n Energy and Climate Change Research Group, University of Southampton, Highfield, Southampton SO17 1BJ, UK
a r t i c l e in f o
abstract
Article history: Received 10 November 2010 Accepted 8 December 2011 Available online 2 February 2012
This paper describes an experimental investigation into the effect of restricting the vortexinduced vibrations of a spring-mounted rigid cylinder by means of stiff mechanical endstops. Cases of both asymmetric and symmetric restraint are investigated. Results show that limiting the amplitude of the vibrations strongly affects the dynamics of the cylinder, particularly when the offset is small. Fluid-structure interaction is profoundly affected, and the well-known modes of vortex shedding observed with a linear elastic system are modified or absent. There is no evidence of lock-in, and the dominant impact frequency corresponds to a constant Strouhal number of 0.18. The presence of an end-stop on one side of the motion can lead to large increases in displacements in the opposite direction. Attention is also given to the nature of the developing chaotic motion, and to impact velocities, which in single-sided impacts approach the maximum velocity of a cylinder with linear compliance undergoing VIV at lock-in. With symmetrical end-stops, impact velocities were about one-half of this. Lift coefficients are computed from an analysis of the cylinder motion between impacts. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Fluid-structure interaction Vortex-induced vibration Nonlinear restraints Impacts
1. Introduction Vortex-induced vibrations (VIV) of a rigid circular cylinder on linear elastic supports, free to move in the direction transverse to the flow, have been thoroughly investigated in the past. This is a canonical problem relevant to much more complex systems, such as long flexible risers used in the offshore industry. Several authoritative reviews exist, e.g. Bearman (1984, 2011), Sarpkaya (2004) and Williamson and Govardhan (2004). The essentials are well known. Flow separation from a cylinder (diameter D) in a steady fluid flow (velocity U) may result in the alternate shedding of vortices behind the structure. The resulting unsteady force on the cylinder can lead to vibrations that form part of a complicated mechanism of fluid-structure interaction. In the lock-in range, the dominant frequency is strongly influenced by the cylinder natural frequency (fn) in fluid initially at rest, while in other conditions it may be close to the frequency of vortex shedding (fvs) from a stationary cylinder. This latter frequency defines a Strouhal number St¼fvsD/U that is approximately 0.2 over a wide range of Reynolds numbers. The range of flow speeds at which lock-in occurs, the structure of the wake and typical amplitudes of response, are all also strongly dependent on the mass ratio mn ¼ ms/md where ms is the mass of the cylinder and md ¼ r pD2/4 is the displaced mass, both per unit length, and on the structural damping (z, as a proportion of critical damping). This paper is concerned with VIV of a cylinder with low mass ratio and low structural damping, in the presence of stiff mechanical end-stops that restrict its displacement either on one side, or on both sides, of its initial position. The work was
n
Corresponding author. Tel.: þ 44 23 8059 2843. E-mail addresses: [email protected] (S. Bourdier), [email protected] (J.R. Chaplin). 1 Present address: Marine Renewables Research Group, Queen’s University Belfast, Belfast BT9 5AG, UK.
0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfluidstructs.2011.12.014
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motivated by interest in the general problem of nonlinear restraints imposed on a complex mechanism of flow-induced vibration, a situation that arises in some industrial applications. Examples, covering a range from mild to severe nonlinearities, include spars with mooring systems that exhibit nonlinearities, catenary risers that intermittently touch down on the seabed, hybrid risers that incorporate flowlines passing through loose fitting spacers and adjacent conductors that experience clashing. The configuration that is addressed in the present paper is not directly applicable to any of these, but represents an extreme case that nevertheless has the advantage of being simply-defined, and therefore more likely to provide some physical insights. Previous experiments on VIV of a cylinder with nonlinear compliance include those by Hover and Triantafyllou (1998) and Stappenbelt and Thiagarajan (2004). In both, the support hardening elastic system had a cubic term that raised the stiffness by up to about 50% at a displacement of one diameter. Neither paper reports dramatic changes in the cylinder response, relative to the purely linear case. Hover and Triantafyllou noted that the lock-in range was narrower, and Stappenbelt and Thiagarajan observed reductions in peak amplitudes by up to 18% in the lock-in range, and a reduction in the corresponding flow velocity. More violent perturbations to VIV responses, and structural damage, can be expected where hard surfaces are impacted at high velocity, as in riser clashing (Sagatun et al., 2002). Numerical simulations of the vortex-induced vibrations of a cylinder by Zhao and Cheng (2011) included cases in which the cylinder bounced back after contact on a plane boundary. Another strand of the literature is concerned with vibration-induced impacts of closely-spaced heat exchanger tubes that pass at intervals through loosely-fitting supports. Reviews include Gelbe et al. (1995) and Ibrahim (2011). Vortex shedding is one of a number of excitation mechanisms that occur in this situation, and its importance relative to that of the potential flow forces is discussed by Kevlahan (2011). But owing to the proximity of neighbouring tubes it does not have much in common with that of an isolated cylinder in a steady cross-flow. Nevertheless it is worth noting that, for example, Mureithi et al. (1994a,b) observed a sequence of bifurcations, including period-doubling, flip and pitchfork bifurcations in tube responses, as the flow velocity was increased, leading to chaotic motion. Some of these phenomena may occur also when the VIV of an isolated cylinder is in some way constrained. This paper presents measurements of the VIV of a cylinder in three configurations. After a description of the experimental setup in Section 2, it is shown that the results of tests with a purely linear elastic support system are in close agreement with published data. The response of the cylinder in the presence of end-stops placed first on one side, and then symmetrically on both sides, is discussed in Sections 3 and 4. Depending on the offset and the reduced velocity, the cylinder may exhibit periodic or chaotic motion, with corresponding changes in the magnitude and frequency of impacts. Large displacements in the opposite direction may occur when there is only one end-stop present. Section 5 is concerned with the calculation of lift coefficients from the measurements, and the problem of predicting the motion if the forcing were known. 2. Experimental details 2.1. Description of the apparatus To obtain low turbulence conditions, these experiments were carried out in a towing tank, which was 17 m long, 0.42 m wide, with a water depth of 0.7 m. The design of the remainder of the apparatus was aimed at achieving a low mass ratio and low structural damping. The cylinder, a sealed acrylic tube, had an outer diameter D of 30 mm, a wall thickness of 3 mm and a length of 300 mm between 2.5D diameter endplates. It was mounted horizontally on the elastic system sketched in Fig. 1, and when stationary its axis was 5.5D beneath the free surface. As can be seen, the cylinder was attached beneath a vertical frame which was connected to one end of each of three parallel beams (two towards the bottom and one at the top), mounted at the other end on the carriage. The beams (aluminium strips 1.5 mm 15 mm 920 mm—from which the vertical frame supporting the cylinder was also constructed) remained above the water surface at all times. The system was balanced, making the beams horizontal when the rig was stationary, by adjusting tension springs that connected the top of the vertical frame to the carriage. A photograph of the rig on the carriage is shown in Fig. 2. The frequency of the first mode of oscillation of the system (shown in Fig. 1), with the cylinder in still water, was measured at 0.977 Hz. Subsequent modes and modal frequencies were computed by means of finite elements. The next two, at 2.7 Hz and 8.1 Hz, are torsional modes involving axial oscillations of the cylinder, that would be heavily damped by its endplates. The fourth and fifth modes (8.4 Hz and 9.4 Hz) involve primarily flexing of the beams within their length, and remaining modal frequencies are all above 10 Hz. There was no evidence of any rocking motion in any of the tests. For a vertical displacement of one diameter (30 mm) the horizontal displacement of the cylinder in the first mode of oscillation was less than 0.6 mm. The calculated fundamental natural frequency of the submerged cylinder between pinends was 150 Hz. In the experiments described here, the mass ratio mn (including the effect of contributions from all moving parts) was 1.74. With carriage speeds U between 0.1 m/s and 0.45 m/s the corresponding range of reduced velocities Ur ¼U/fnD was from about 3 to 15. The structural damping in air was 0.2% of critical damping. In water, additional system damping was generated by viscous shear stresses acting on the cylinder’s end plates and on immersed parts of the vertical beams supporting it. Measuring this contribution would entail replacing the cylinder by a horizontal tie bar, adding ballast to represent its inertia and adjusting the balancing springs to return the support system to the same zero
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Fig. 1. Sketch of the cylinder support system showing a side view on the left, and a section of the tank on the right. The mounting points for the beams, the balancing springs and the position sensor were fixed on the carriage, which towed the whole apparatus along the tank at constant speed. Dashed lines show the deflected shapes of the beams in the first mode of oscillation for an amplitude of one diameter.
Fig. 2. Apparatus on the carriage.
position—alterations that would themselves inevitably have an effect on the damping. An alternative approach, found in earlier experiments to provide a good estimate of low levels of hydrodynamic damping (Chaplin, 2000), was to use Stokes’s pffiffiffiffiffiffiffiffiffiffiffiffi formula t ¼ ru on=2 for the shear stress on a plate oscillating with velocity u in its own plane in an unbounded fluid. At a representative frequency of 1 Hz, this led to an estimate of the hydrodynamic damping on all submerged components (except the cylinder itself) of 0.59% of critical damping, suggesting that the effective structural damping z was about 0.79%, or mnz ¼0.014. The system stiffness k1 in this unmodified configuration was measured at 65 N/m per metre length of cylinder. The vertical displacement of the cylinder was recorded by means of a Hamamatsu C2399 optical position sensor system consisting of an active infrared LED mounted on the oscillating frame, and a camera on the carriage. 2.2. Validation of the response in the case of linear structural restraints Before installing end-stops to limit the motion, measurements of VIV were checked against published results. Representative time series of the cylinder displacement y are plotted in Fig. 3, with the corresponding spectra alongside. Peak amplitudes and frequencies are shown in Fig. 4, with data from Khalak and Williamson (1997) for mn ¼2.4 (somewhat higher than the present value of 1.74), and mnz ¼0.014 (as in the present case). In Fig. 4 Anmax is the maximum amplitude of response divided by the cylinder diameter D, and values of fn denote dominant spectral frequencies divided by its natural
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Fig. 3. On the left, displacement time series for vibrations without end-stops, on the right, corresponding spectra.
frequency in still water fn. The results are generally in good agreement with expectations, displaying the initial, upper and lower branches that are characteristic features of the VIV response in these conditions. In the first part of the initial branch, two peaks appear in the spectrum (Fig. 3(a)), at fn (the cylinder natural frequency in still water) and at a frequency denoted fvs. At slightly higher reduced velocities (Fig. 3(b)), the motion is nearly periodic, and there is only one significant peak in the spectrum, at fvs, with Anmax o0:4. The upper branch starts at Ur ¼4.78 when the frequency ratio fnof the periodic motion shown in Fig. 3(c) jumps to unity, and the maximum amplitude jumps to about 0.8D. The motion remains periodic in the first part of the upper branch and the dominant frequency increases slightly with Ur, a characteristic feature of VIV with low mass ratios and low damping (Khalak and Williamson, 1997). n At Ur ¼7.16, a second peak at a frequency denoted f lower appears in the frequency distribution, revealing the start of the transition between upper and lower branches. Over this transition, the cylinder switches intermittently between the two modes, as shown by the motion in Fig. 3(d) for Ur ¼7.68. With increasing Ur, the proportion of time spent by the cylinder in the upper branch mode of vibration diminishes until only the lower mode of oscillation is evident, at Ur ¼8.88 (Fig. 3(e)). In the subsequent lower branch the motion is periodic, with an amplitude and frequency remarkably constant, with n An E0.55 and fn ¼ 1.48, close to the value of 1.51 predicted by the formula for f lower proposed by Khalak and Williamson n (1999) for m ¼1.74 and an added mass coefficient Ca ¼ 1. The reduced velocity where desynchronisation occurs is difficult to identify, but a peak at fvs appears in the frequency distributions at Ur ¼11.44. For higher velocities, the motion becomes much more complex and disorganised, leading to the presence of several peaks in the much wider frequency spectra, including those at fn, fvs and flower. Instabilities can sometimes give rise to a few high amplitude cycles, like those seen in Fig. 3(f) for Ur ¼ 12.63 at tE10 s. The measured values of fvs correspond to a Strouhal number St ¼0.18 which is rather lower than Khalak and Williamson’s (1997) result of 0.22, but close to that in other experiments, e.g. those for a smooth cylinder by Hover et al. (2001) at mnz ¼0.096. Overall, the measurements of VIV obtained with this linear experimental setup show good agreement with comparable results in the literature. It exhibited exactly the characteristic low-mass ratio and low-damping amplitude and frequency responses expected in such cases.
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Fig. 4. (a) Non-dimensional peak frequencies fn and (b) maximum amplitudes Anmax , plotted as functions of reduced velocity for vibrations without endstops: ’, present results with mn ¼ 1.74, mnz ¼ 0.014; J, data from Khalak and Williamson (1997) with mn ¼2.4, mnz ¼ 0.014. The solid line in (a) represents a Strouhal number of 0.22, the broken line, 0.18.
3. VIV with impacts on a single end-stop 3.1. Experimental arrangements Following the validation tests, an end-stop was added to the system to limit the motion on one side, at an offset y/D ¼ eo 0. The end-stop, a short steel bar mounted on the carriage, did not limit the motion of the cylinder directly, but restricted the displacement of its supporting frame above, as can be seen in Fig. 1 (where two symmetrical end-stops are shown). Owing to flexibility in the frame, the effect of contacting an end-stop was therefore that of a sudden increase in structural stiffness rather than a rigid limit. This increased stiffness k1 þk2 was measured at approximately 22.3 kN/m per metre length of cylinder. This is illustrated in Fig. 5, where the nonlinear cases studied by Hover and Triantafyllou (1998) and Stappenbelt and Thiagarajan (2004) are shown as broken lines. In the present experiments three series of tests were carried out, with the end-stop placed in turn at e¼ 0.65 (shown), 0.46 and 0.22.
3.2. Modification of the cylinder dynamics Representative time series of displacements are shown in Fig. 6(a)–(d) for Ur ¼5.46. For the linear elastic case (Fig. 6(a)) this is within the upper branch where the cylinder exhibits a periodic oscillation with amplitude of about 0.8D. When an end-stop is placed at 0.65D the cylinder impacts on it at every cycle, but the result is a stable motion with little amplitude modulation. The same applies at e¼ 0.46, and in both these cases the peak excursion of the cylinder on the side away from the end-stop is reduced to a displacement that almost mirrors the position of the end-stop. At e¼ 0.22 however there is a dramatic change, and the response becomes much more complicated. Through this sequence there is a slight reduction in the dominant frequency from fn ¼1.08 for the unrestricted case, to fn ¼0.99 (St ¼0.18) for e¼ 0.22. At a reduced velocity of 10.9 (Fig. 6(e) and (f)), the unrestricted motion is in the lower branch, and has an almost constant amplitude of about 0.5D. The effect of an end-stop at e¼ 0.22, is to produce a motion with strongly modulated amplitudes, and peak excursions that extend much further, to 1.6D, on the side away from the end-stop. There is also a marked increase in the dominant frequency from fn ¼1.51 (in the unrestricted case) to fn ¼1.85 (St¼ 0.17). Bifurcations diagrams (Fig. 7) reveal the ranges of reduced velocities over which impacts occur, and confirm as expected that virtually all changes in direction from negative to positive velocities take place in the region yo0. They show the stability of 1-impact-per-period motions over most of the impacting domain in all cases. Windows of chaotic 1-impact-per k cycles motions, in which the cylinder describes k cycles of oscillation between one impact and the next, with k varying widely, occur at the start of each impacting domain. For e¼ 0.65, this window is very narrow. It is wider for e¼ 0.46,
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Fig. 5. The elastic resistance of the system, with (a) one end-stop and (b) two end-stops shown as continuous lines. The system characteristics in experiments by Hover and Triantafyllou (1998) and Stappenbelt and Thiagarajan (2004) are shown as broken lines, scaled to match the inner stiffness in the present experiments.
Fig. 6. On the left, time series of dimensionless displacements recorded at Ur ¼ 5.46, (a)–(d), and at Ur ¼10.9, (e)and (f), with the positions of the single end-stop (when present) as indicated. Corresponding spectra are plotted on the right.
and wider still for e¼ 0.22, extending up to Ur ¼7. Windows of chaotic impacting motions can be observed for e¼ 0.46 for Ur 410 and for e¼–0.22 for Ur 412. Fig. 7(c) includes measurements of VIV amplitudes by Stappenbelt and Thiagarajan (2004) with linear and nonlinear elastic supports. Their mass-damping ratio was considerably higher than in the present experiments, and even with the linear system, displacements did not exceed An ¼0.45. Nevertheless, in both experiments, amplitudes observed with nonlinear restraints reached a maximum at reduced velocities higher than in the linear case.
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Fig. 7. Bifurcation diagrams of peak excursions in both directions for single-sided impact cases: (a) e¼ 0.65; (b) e¼ 0.46; (c) e ¼ 0.22. In each plot, results for the linear elastic system are shown as grey symbols. Plot (c) includes measurements of response amplitudes by Stappenbelt and Thiagarajan (2004) with mnz ¼0.222 (considerably higher than in the present experiments): linear system and &, their most nonlinear system (see Fig. 5).
Fig. 8. Peak frequencies of single-sided impacting motions. The line represents a Strouhal number of 0.18.
Peak frequencies in the response for impacting motions at all offsets are plotted in Fig. 8. Many points lie close to the line representing a Strouhal number of 0.18, though in these conditions for the smallest offset there was a wider bandwidth, and in some cases two or even three distinct dominant frequencies. An example appears in Fig. 6(f), reflecting the beating motion that can be seen in the time series. The perturbation of the dynamics of the cylinder caused by the presence of an end-stop can also be observed by means of phase portraits, plotting the trajectory of the system in the plane _ c Þ, as shown in Fig. 9. Many of these features described above can be seen here also. ðy=D, y=U 4. VIV with impacts on two symmetrically-placed end-stops Four series of tests were conducted with end-stops at e¼ 70.20, 70.30, 70.47 and 70.65. For e¼ 70.65, impacts occurred only over the range of reduced velocities corresponding to the upper branch in the linear case. For e¼ 70.47, it occurred over the upper and lower branches, and for e¼ 70.30 and 70.20 it occurred over nearly all the lock-in domain. Representative time series of dimensionless displacements y/D recorded at Ur ¼5.12 are plotted in Fig. 10. For e¼ 70.65 and 70.47, the presence of the end-stops does not seem to affect the periodic nature of the motion, and with very few exceptions the cylinder impacts once on each stop during each cycle of oscillation.
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_ c axes have the same Fig. 9. Phase portraits for the linear elastic case and for single-sided impacting motions. All y/D axes have the same scale, and all y=U scale. The motion is clockwise around each trajectory.
When the end-stops are moved in, in this case to e¼ 70.30 or 70.20, the cylinder displacement exhibits sequences of symmetric double-impact motions, but also cases where it impacts successively on the same stop or where it undergoes an oscillation near the mid-point. The motion in these cases is much more affected by the end-stops and the response is more complex without any distinctive periodic pattern. The frequency content of the cylinder displacement also shows the perturbation of the dynamics caused by the presence of the end-stops. As illustrated by the frequency ratio distributions presented in Fig. 11, the magnitude of the main peak diminishes as e decreases, while those in the side bands increase. In Fig. 11, frequency distributions for the linear elastic case are presented in the first column, while those in the other columns correspond to double-sided impacting motions with the offset decreasing while going to the right. The reduced velocity increases when going down a row. The energy becomes more widely distributed as the motion is more constricted. Frequency distributions for motions with small end-stop offsets, exhibiting some structure but with energy distributed over wide ranges of frequencies, provide an indication of chaotic motions. Even though the motion appears to be substantially unaffected by impacts on end-stops at e¼ 70.65 in Fig. 10, for most reduced velocities its frequency was not the same as in the linear elastic case. In Fig. 12 the peak frequency is plotted as a function of Ur and in fact is seen to rise at a rate corresponding to a constant Strouhal number, without lock-in, for all double-sided impacting cases over their respective impacting domains. The absence of lock-in is consistent with the results of Hover and Triantafyllou (1998), in which (with weaker nonlinearities—see Fig. 5), the lock-in range was considerably narrower than in the linear case. Characteristic motions seen in the phase portraits in Fig. 13 can be classified using the identification k-‘-m where in k cycles the cylinder impacts ‘ times on the upper stop and m times on the lower stop. For e¼ 70.65 at the reduced velocities shown, the cylinder exhibits a stable 1-1-1 motion. When the offset is reduced below e¼ 70.65, the trajectories become less repeatable. This trend continues as e is reduced, and at e¼ 70.30 and 70.20 there is barely any discernable pattern and the trajectories fill most of the phase plane enclosed by their widest and narrowest cycles, again indicative of a motion of a chaotic nature. Nevertheless some characteristic features can be identified. At low reduced velocities the cylinder amplitude is strongly decreased after an impact and the cylinder may describe several cycles of amplitude growth before its next impact. An example of such a sequence, taken from the test with Ur ¼4.44 and e¼ 70.20, is shown in Fig. 14(a), where the cylinder can be observed describing a k-0-1 motion with k ¼4, i.e. four oscillation cycles after leaving the lower stop before reaching the lower stop again. Under these conditions k varied between 2 and 10 without any apparent pattern. The maximum period of these motions, largest after the first impacts, decreases as Ur increases. In Fig. 14(b) (Ur ¼4.78) the motion is 2-0-1, while in Fig. 14(c) (Ur ¼7.51) the cylinder follows successive anti-symmetric 1-1-1 cycles, the second one shown as a dashed line. Also, trajectories with successive bounces on the same end-stop occurred for cases with small offsets and high reduced velocities. Examples are shown in Fig. 14(d) (Ur ¼10.92) and in Fig. 15 (Ur ¼11.95), where it can be seen that the cylinder sometimes chattered against the end-stops for as long as one complete period of end-to-end oscillation. In those cases in which a distinct frequency can be detected in the chattering motion, it is above 30 Hz.
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Fig. 10. Displacement time series at Ur ¼ 5.12: (a) for the linear elastic case, and for double-sided impacting motions at (b) e ¼ 70.65; (c) e¼ 7 0.47; (d) e¼ 70.30; (e) e¼ 7 0.20.
Fig. 11. Frequency distributions for the linear elastic case (on the left) and for double-sided impacting motions.
Bifurcation diagrams of lower peak excursions, showing points on the (Ur,y/D) plane at which, in individual cycles, the trajectory crossed the Poincare´ section ðy, y_ ¼ 0, tÞ from y_ o 0 to y_ 4 0 are shown in Fig. 16. In these diagrams the points show impacts on the lower stop as well as positions between the stops at which the cylinder reversed direction from down
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Fig. 12. Peak frequencies in double-sided impacting motions. The line represents St ¼0.18.
Fig. 13. Phase portraits for linear restraints and double-sided impacting motions. Broken lines represent the positions of the end-stops. The motion is clockwise around each trajectory.
Fig. 14. Phase portraits taken from the motions at e ¼ 7 0.20 and reduced velocities Ur of (a) 4.44, (b) 4.78, (c) 7.51 and (d) 10.92.
to up. For e¼ 7 0.65 the bifurcation diagram shows the upper branch being transformed into a window of stability of the 1-1-1 motion, since reversals take place only at the stop position. The 1-1-1 motion is also dominant for e¼ 70.47 over the entire impacting domain, even if some lower peak excursions scattered over the negative part of the bifurcation diagram reveal some loss of stability. When e is decreased to 70.30, numerous lower peak excursions scattered over the negative and positive sides of the bifurcation diagram reveal the complex nature of the motions in this case. The increased density of points close to the upper stop position at the end of the impacting domain, for Ur 410, shows the occurrence of successive impacts on the upper stop, or 1-‘-0 trajectories with ‘ 4 1. For the lowest offset value, lower peak excursions are densely spread across the bifurcation diagram, revealing the complex nature of the motion over the impacting domain in this case. Successive impacts on the upper stop or chattering
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Fig. 15. Displacement time series for e¼ 7 0.20, Ur ¼11.95.
Fig. 16. Bifurcation diagrams of lower peak excursions for double-sided impacting motions. (a) e ¼ 7 0.65, (b) e¼ 70.47, (c) e¼ 7 0.30 and (d) e¼ 70.20. Symbols in grey refer to the case without end-stops.
motions start to occur in this case much earlier than at e¼ 70.30, at around Ur ¼8. It is interesting to note that, once started, impacts continue to occur on end-stops placed beyond the peak excursion of the cylinder in the case without endstops (e.g. in Fig. 16(b)).
5. Lift coefficients 5.1. A single end-stop When the cylinder encounters or leaves an end-stop it experiences a sudden large change in structural stiffness. The subsequent motion (until the cylinder next leaves or meets an end-stop) is that of a forced single degree of freedom system, released from a defined starting position with a defined velocity. In such a case, if the forcing F(t)were known (where t is measured from the instant at which the change in stiffness takes place), the motion y(t) up to the time of the next impact could be found by means of the Laplace transform of its equation of motion pffiffiffiffiffiffiffi my€ þ2z mky_ þky ¼ FðtÞ, ð1Þ
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using the known initial displacement and velocity as the two required boundary conditions. For present purposes it seems reasonable to represent the fluid forces in the form F ¼ 12rU 2 d½C Y0 þC Y cosðotaÞ,
ð2Þ
in which a time-independent component (represented by CY0) might appear because of the asymmetric nature of the motion when only one end-stop is present. The solution of (1) subject to forcing of the form of (2) with y¼y0 and y_ ¼ y_ 0 at time t ¼0 is given in the appendix. Apart from the time independent term it consists of apcomponent of constant amplitude ffiffiffiffiffiffiffiffiffiffi at the driving frequency o, and a decaying component at the natural frequency o0 ¼ k=m. If the structural damping is light, it has little effect on the initial motion (say over a time interval of 2p/o), which can then be approximated very closely by putting z ¼0 in Eq. (A2). This is then simply the sum of two sinusoids and a constant term. Matching this formula to the observed displacement of the cylinder between end-stop contacts provides a way of estimating lift coefficients for the impacting motions described above. In the case of a cylinder undergoing VIV on elastic supports without restraints, lift coefficients can be computed from displacement records by following the approach used by Khalak and Williamson (1999). This consists basically of reorganising Eq. (1) (with m representing the effective mass of the cylinder) and (2) (without CY0), to find those values of CY cos a and CY sin a such that both the peak displacement, and the frequency indicated by the sum of the system’s inertia and the force in anti-phase with the acceleration (represented by CY cos a), match the measurements. In that situation the choice of the meaning of m in Eq. (1) is largely a matter of convention, since changing to any other definition would be accommodated by an adjustment in the value of CY cos a. In the present case, when trying to match the measured response to Eqs. (2), this freedom does not exist, because the response is likely to contain a distinct component at frequency p(1) ffiffiffiffiffiffiffiffiffiand ffi o0 ¼ k=m. Therefore m in Eq. (1) should include the effective added mass, which is initially unknown. To analyse the present data, a least squares procedure was applied over each interval between the cylinder leaving the end-stop and its next impact, with the aim of minimising the root-mean-square difference E between the measured displacements over the duration of a complete run, and the motion obtained from Eq. (A2) on a cycle-by-cycle basis. For a given value of the added mass coefficient Ca (making m¼md(mn þCa) where md is the displaced mass) the variables to be evaluated in the least squares analysis are the forces F 0 ¼ 12rU 2 C Y0 , F C ¼ 12rU 2 C Y cos a and F S ¼ 12rU 2 C Y sin a. The driving frequency o in each case was taken to be that which corresponds to a Strouhal number of 0.18, matching the trends seen in Figs. 8 and 12. It might be expected that a least squares approach could deliver also Ca for each run, but the system of equations is not sufficiently well conditioned to provide robust solutions with four unknowns. Instead, a constant value was adopted throughout. This was based on results like those in Fig. 17, in which the root-mean-square error (with best-fit values for the other variables) for each of six cases is plotted as a function of the trial value of Ca. The average position of the minimum error over all cases was Ca ¼0.65, and this was used in subsequent calculations. Displacements, computed from Eq. (A2) with the amplitude of each harmonic component determined by least-squares analysis, are generally in excellent agreement with the measured data, as shown in Fig. 18. In most cases the measurements (plotted as continuous lines) are practically indistinguishable from the least-squares fit (broken lines), except where there are several oscillations between one impact and the next (e.g. in Fig. 18(d) and (e)). It seems reasonable to assume that this is because the vortex-induced forces are strongly modified when the cylinder is no longer undergoing sudden reversals at frequent intervals. Mean lift coefficients computed from the least-squares fit are plotted in Fig. 19. A marked difference from the case of unrestrained VIV (Fig. 19(a)) is the absence of high values of CY at Ur E4.7. This is normally the beginning of the upper branch of the response, but as noted above that does not appear when end-stops are limiting the motion. On the other hand, lift coefficients for impacting motions tend to rise at higher reduced velocities, unlike those in the case of VIV with linear restraints. The mean force, normalised as CY0, was extremely small in all cases. In the case of VIV with linear restraints, if one were to integrate the cylinder’s equation of motion (1) using experimentally obtained force coefficients for the forcing on the right hand side, it would be reasonable to expect that the
Fig. 17. Root-mean-square errors in predictions of displacements obtained with Eq. (A2).
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Fig. 18. Displacement time series: continuous lines are measurements, broken lines from Eq. (A2) with optimal coefficients and Ca ¼ 0.65. Each impact on the end-stop is identified with a symbol. (a) e¼ 0.65, Ur¼5, (b) e¼ 0.65, Ur¼8.0, (c) e¼ 0.46, Ur¼5.1, (d) e¼ 0.46, Ur¼11.1, (e) e¼ 0.22, Ur¼5.5 and (f) e¼ 0.22, Ur¼11.9.
computed motion would closely match the measurements, at least in those conditions where the motion is strongly periodic. Moreover, in such cases, where the force coefficients are smooth functions of the reduced velocity, the same would apply to computed amplitudes. This is quite different from the behaviour of an impact or bi-linear oscillator responding to harmonic forcing. Much of the literature on that subject is concerned with the consequences of the extreme sensitivity of such systems to the controlling parameters, and the presence of bifurcations in their response (e.g. Foale and Bishop, 1994; Natsiavas, 1990). Response amplitudes obtained by numerically integrating the equation of motion of a system that incorporates sudden changes in stiffness like those in Fig. 5 do not in general exhibit the smooth qualities of the linear case. Small changes in the amplitude or frequency of the forcing, or in the starting conditions, may cause the motion to jump from one type of response to another. To check how well the displacements of the cylinder measured in the present experiments could be recovered from a knowledge of the characteristics of the elastic system and the best-fit lift coefficients, a Newmark-b scheme (with b ¼1/6 and g ¼1/2) was used to integrate the cylinder’s equation of motion, with a bisection routine to identify the times of contact with, and separation from, the end-stop, and an appropriate change in stiffness at each transition. Test runs were carried out to validate the code, by comparing results with those obtained from the analytical solution for the same problem developed by Bapat (2008, 2011). Typical computations advanced in as many as 10 000 time steps in every cycle
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Fig. 19. Mean lift coefficients for the case of no end-stops, and for single-sided impacting motions. In (a), data from Khalak and Williamson (1997) are shown with unfilled symbols. (a) No end-stops, (b) e¼ 0.65, (c) e¼ 0.46 and (d) e¼ 0.22.
Fig. 20. Average impact velocities on a single end-stop: þ , measurements; , results of a numerical time-stepping model of the motion of the cylinder, computed using lift coefficients provided by the polynomials plotted in Fig. 19 with dashed lines. (a) e¼ 0.65, (b) e¼ 0.46 and (c) e¼ 0.22.
of the forcing, and reached a steady state before 1000 cycles had been completed. The process was started with the system at rest at zero offset, and the forcing ramped up over 20 cycles. Numerical results agreed with those obtained from Bapat’s analytical approach to at least 5 decimal places. The same programme was used to model the response of the cylinder at each of a large number of reduced velocities. In each case the frequency of the forcing corresponded to a Strouhal number of 0.18, and its amplitude was derived from a polynomial (shown as a dashed line for each end-stop position in Fig. 19) fitted to the measured lift coefficients. Some
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Fig. 21. Average impact velocities on two symmetrical end-stops: þ , measurements; , results of the numerical time-stepping impact oscillator, using the same lift coefficients as in Fig. 20, i.e. those based on the measurements of single-sided impacts. (a) e¼ 0.65, (b) e¼ 0.47 and (c) e¼ 0.20.
results are plotted in Fig. 20, in the form of the mean velocity of the cylinder at the instant of impact on the end-stop, normalised with respect to Uc ¼2pfn/D. Numerical results are shown as solid grey circles. As expected, over some ranges of reduced velocity, there are bifurcations in the computed response. There are also regions, Ur 46.8 at e¼ 0.65, Ur 47.9 at e¼ 0.46 and 9.7oUr o11.4 at e¼ 0.22, where no impacts occur. When seen in this light, jumps from one branch to another can be detected also in the measured impact velocities (plotted in Fig. 20), at about Ur ¼ 5.5 for all three end-stop positions. In most cases the velocities are in reasonable agreement with the numerical results, though impacts occurred in conditions where in the numerical model there was no contact. Differences between measured and predicted velocities may be associated with the absence, in the numerical modelling, of any feedback from the cylinder motion to the forcing. It seems reasonable to assume that the very high acceleration of the cylinder that occurs during every impact should lead to a departure, at least momentarily, from a purely harmonic lift force. It would be necessary to model the flow in order to include this effect in numerical predictions. 5.2. Symmetrical end-stops Over the short interval between impacts on symmetrically placed end-stops, there was not enough information from the cylinder trajectory to support the calculation of force coefficients as attempted above. Because the motion was more restricted, impact velocities were lower than in the case of a single end-stop, and are plotted in Fig. 21. They are compared there with predictions of the numerical impact oscillator model, using the same lift coefficients as before. Again there is some measure of agreement, but flow modelling would be required to achieve a better match. 6. Conclusions Measurements have been carried out of the vortex-induced cross-flow vibrations of an isolated stiff cylinder mounted on an elastic system in the presence of end-stops that introduced a very large increase in the stiffness of the system at a particular offset, either on one side or on both sides of the cylinder’s initial position. The mass ratio was 1.74 and structural damping was 0.79% of critical. In the absence of the end-stops, measurements of VIV were shown to be in close agreement with previous data for systems with similarly low mass ratio and low damping. Placing end-stops at an offset of 0.65 diameters, on one or both sides, eliminated any evidence of frequency lock-in. The dominant frequency in the response remained close to that indicated by a constant Strouhal number of 0.18 over the range of reduced velocities between about 4 and 14, and the motion was characterised by a single impact on each end-stop
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in every period. More complicated displacement time histories appeared with offsets at 0.3 diameters or less, when the cylinder exhibited successive impacts on the same end-stop, reversals near its mean position, and chattering on one of the end-stops for as long as one complete period of oscillation. Response spectra reveal a progressive growth in sub- and superharmonics, as well as a general broadening of the frequency content. When there were regular impacts on a single end-stop, the motion of the cylinder between impacts was shown to match the expected trajectory of a forced linear single-degree-of-freedom system released from an initial offset with a given velocity. It was then possible to extract lift coefficients and compare measured impact velocities with those computed from a numerical model of an impact oscillator, using the same forcing. The results were rather mixed, but illuminated the existence of jumps in the measurements between modes of response. These and other features of the observed response are indicative of a degree of chaos that increases as the motion of the cylinder is more and more confined, in conditions that give rise to lock-in in the linear elastic case. This was confirmed by computations of the Lyapunov exponent that are beyond the scope of the present paper but are reported in Bourdier (2008). At reduced velocities around 10, the motion of the cylinder when constrained on both sides appeared more chaotic than when a single end-stop was present, as can be seen in bifurcation diagrams. In the latter case the cylinder impacted on the end-stop once in every cycle, but rebounded with an excursion of as much as one and a half diameters beyond its stationary position in the opposite direction. In both cases mean impact velocities increased with the reduced velocity, and maximum mean values were close to Uc ¼2pfn/D, or in other words comparable to the maximum velocity of a cylinder undergoing lock-in VIV with linear elastic compliance. Further work could usefully be carried out on the effect of other forms of nonlinearities in the support system, and, in the present configuration, on the influence of the stiffness of the end-stops. A companion paper (Bourdier and Chaplin, in preparation) compares the present measurements with numerical predictions based on wake oscillator models. Appendix A Consider a single-degree-of-freedom system pffiffiffiffiffiffiffi my€ þ 2z mky_ þ ky ¼ F 0 þ F c cos ot þF s sin ot,
ðA1Þ
released at time t ¼0, from a position y¼ y0 with a velocity y_ ¼ y_ 0 . Then by means of a Laplace transform of Eq. (A1) it can be shown that the ensuing motion is y¼
F 0 ð1O2 ÞF c 2zOF s 2F c Oz þ ð1O2 ÞF s SF 0 þ ð1O2 ÞF c 2zOF s kSy0 zo0 t 0 þ cos ot þ sin ot e cos z o0 t k kS kS kS
zSF 0 þ zð1O2 ÞF c þ Oð12z2 O2 ÞF s Sðkzy0 þmo0 y_ 0 Þ zo0 t 0 e sin z o0 t, z0 kS
ðA2Þ
qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 0 2 where o0 ¼ k=m, O ¼ o/o0, z ¼ 1z and 2
S ¼ 12O2 þ4O2 z þ O4 :
ðA3Þ
It can be seen that the response consists of a steady offset, a term of constant amplitude at the driving frequency o and a term of decaying amplitude at frequency o0. References Bapat, C.N., 2008. Exact solution of stable periodic one contact per N cycles motion of a damped linear oscillator contacting a unilateral elastic stop. Journal of Sound and Vibration 314, 803–820. Bapat, C.N., 2011. Correction of Misprints in Bapat (2008). Personal Communication. Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics 16, 195–222. Bearman, P.W., 2011. Circular cylinder wakes and vortex-induced vibrations. Journal of Fluids and Structures 27, 648–658. Bourdier, S., 2008. Vortex-Induced Vibrations of a Nonlinearly Supported Rigid Cylinder. Ph.D. Thesis. University of Southampton, UK. Bourdier, S., Chaplin, J.R. Vortex-induced vibrations of a rigid cylinder on elastic supports with end-stops. Part 2: numerical modelling and comparison with measurements, in preparation. Chaplin, J.R., 2000. Hydrodynamic damping of a cylinder at b E106. Journal of Fluids and Structures 14, 1101–1117. Foale, S., Bishop., S.R., 1994. Bifurcations in impact oscillations. Nonlinear Dynamics 6 (3), 285–299. Gelbe, H., Jahr, M., Schroder, K., 1995. Flow-induced vibrations in heat-exchanger tube bundles. Chemical Engineering and Processing 34 (3), 289–298. Hover, F.S., Triantafyllou, M.S., 1998. Lock-in phenomena for cylinders with nonlinear compliance. In: Proceedings of FEDSM98, ASME Fluids Engineering Division Summer Meeting. Washington, DC, USA. Hover, F.S., Tvedt, H., Triantafyllou, M.S., 2001. Vortex-induced vibrations of a cylinder with tripping wires. Journal of Fluid Mechanics 448, 175–195. Ibrahim, R.A., 2011. Mechanics of pipes conveying fluids—part II: applications and fluidelastic problems. ASME Journal of Pressure Vessel Technology 133 (2) (article number 024001). Kevlahan, N.K.-R., 2011. The role of vortex wake dynamics in the flow-induced vibration of tube arrays. Journal of Fluids and Structures 27, 829–837. Khalak, A., Williamson, C.H.K., 1997. Fluid forces and dynamics of a hydroelastic structure with very low mass and damping. Journal of Fluids and Structures 11, 973–982. Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping parameter. Journal of Fluids and Structures 13, 813–851. Mureithi, N.W., Price, S.J., Paı¨doussis, M.P., 1994a. The post-Hopf-bifurcation response of a loosely supported cylinder in an array subjected to crossflow—part 1: experimental results. Journal of Fluids and Structures 8, 833–852.
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Mureithi, N.W., Price, S.J., Paı¨doussis, M.P., 1994b. The post-Hopf-bifurcation response of a loosely supported cylinder in an array subjected to crossflow—part 2: theoretical model and comparison with experiments. Journal of Fluids and Structures 8, 853–876. Natsiavas, S., 1990. On the dynamics of oscillators with bi-linear damping and stiffness. International Journal of Non-Linear Mechanics 25 (5), 535–554. Sagatun, S.I., Herfjord, K., Holmas, T., 2002. Dynamic simulation of marine risers moving relative to each other due to vortex and wake effects. Journal of Fluids and Structures 16, 375–390. Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures 19, 389–447. Stappenbelt, B., Thiagarajan, K., 2004. Vortex-induced vibration of catenary moored cylindrical structures. In: Proceedings of OMAE04, 23rd International Conference on Offshore Mechanics and Arctic Engineering. Williamson, C.H.K., Govardhan, R., 2004. Vortex-induced vibrations. Annual Review of Fluid Mechanics 36, 413–455. Zhao, M., Cheng, L., 2011. Numerical simulation of two-degree-of-freedom vortex-induced vibration of a circular cylinder close to a plane boundary. Journal of Fluids and Structures 27, 1097–1110.
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Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Vortex-induced vibrations of a rigid cylinder on elastic supports with end-stops, Part 1: Experimental results Sylvain Bourdier 1, John R. Chaplin n Energy and Climate Change Research Group, University of Southampton, Highfield, Southampton SO17 1BJ, UK
a r t i c l e in f o
abstract
Article history: Received 10 November 2010 Accepted 8 December 2011 Available online 2 February 2012
This paper describes an experimental investigation into the effect of restricting the vortexinduced vibrations of a spring-mounted rigid cylinder by means of stiff mechanical endstops. Cases of both asymmetric and symmetric restraint are investigated. Results show that limiting the amplitude of the vibrations strongly affects the dynamics of the cylinder, particularly when the offset is small. Fluid-structure interaction is profoundly affected, and the well-known modes of vortex shedding observed with a linear elastic system are modified or absent. There is no evidence of lock-in, and the dominant impact frequency corresponds to a constant Strouhal number of 0.18. The presence of an end-stop on one side of the motion can lead to large increases in displacements in the opposite direction. Attention is also given to the nature of the developing chaotic motion, and to impact velocities, which in single-sided impacts approach the maximum velocity of a cylinder with linear compliance undergoing VIV at lock-in. With symmetrical end-stops, impact velocities were about one-half of this. Lift coefficients are computed from an analysis of the cylinder motion between impacts. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Fluid-structure interaction Vortex-induced vibration Nonlinear restraints Impacts
1. Introduction Vortex-induced vibrations (VIV) of a rigid circular cylinder on linear elastic supports, free to move in the direction transverse to the flow, have been thoroughly investigated in the past. This is a canonical problem relevant to much more complex systems, such as long flexible risers used in the offshore industry. Several authoritative reviews exist, e.g. Bearman (1984, 2011), Sarpkaya (2004) and Williamson and Govardhan (2004). The essentials are well known. Flow separation from a cylinder (diameter D) in a steady fluid flow (velocity U) may result in the alternate shedding of vortices behind the structure. The resulting unsteady force on the cylinder can lead to vibrations that form part of a complicated mechanism of fluid-structure interaction. In the lock-in range, the dominant frequency is strongly influenced by the cylinder natural frequency (fn) in fluid initially at rest, while in other conditions it may be close to the frequency of vortex shedding (fvs) from a stationary cylinder. This latter frequency defines a Strouhal number St¼fvsD/U that is approximately 0.2 over a wide range of Reynolds numbers. The range of flow speeds at which lock-in occurs, the structure of the wake and typical amplitudes of response, are all also strongly dependent on the mass ratio mn ¼ ms/md where ms is the mass of the cylinder and md ¼ r pD2/4 is the displaced mass, both per unit length, and on the structural damping (z, as a proportion of critical damping). This paper is concerned with VIV of a cylinder with low mass ratio and low structural damping, in the presence of stiff mechanical end-stops that restrict its displacement either on one side, or on both sides, of its initial position. The work was
n
Corresponding author. Tel.: þ 44 23 8059 2843. E-mail addresses: [email protected] (S. Bourdier), [email protected] (J.R. Chaplin). 1 Present address: Marine Renewables Research Group, Queen’s University Belfast, Belfast BT9 5AG, UK.
0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfluidstructs.2011.12.014
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motivated by interest in the general problem of nonlinear restraints imposed on a complex mechanism of flow-induced vibration, a situation that arises in some industrial applications. Examples, covering a range from mild to severe nonlinearities, include spars with mooring systems that exhibit nonlinearities, catenary risers that intermittently touch down on the seabed, hybrid risers that incorporate flowlines passing through loose fitting spacers and adjacent conductors that experience clashing. The configuration that is addressed in the present paper is not directly applicable to any of these, but represents an extreme case that nevertheless has the advantage of being simply-defined, and therefore more likely to provide some physical insights. Previous experiments on VIV of a cylinder with nonlinear compliance include those by Hover and Triantafyllou (1998) and Stappenbelt and Thiagarajan (2004). In both, the support hardening elastic system had a cubic term that raised the stiffness by up to about 50% at a displacement of one diameter. Neither paper reports dramatic changes in the cylinder response, relative to the purely linear case. Hover and Triantafyllou noted that the lock-in range was narrower, and Stappenbelt and Thiagarajan observed reductions in peak amplitudes by up to 18% in the lock-in range, and a reduction in the corresponding flow velocity. More violent perturbations to VIV responses, and structural damage, can be expected where hard surfaces are impacted at high velocity, as in riser clashing (Sagatun et al., 2002). Numerical simulations of the vortex-induced vibrations of a cylinder by Zhao and Cheng (2011) included cases in which the cylinder bounced back after contact on a plane boundary. Another strand of the literature is concerned with vibration-induced impacts of closely-spaced heat exchanger tubes that pass at intervals through loosely-fitting supports. Reviews include Gelbe et al. (1995) and Ibrahim (2011). Vortex shedding is one of a number of excitation mechanisms that occur in this situation, and its importance relative to that of the potential flow forces is discussed by Kevlahan (2011). But owing to the proximity of neighbouring tubes it does not have much in common with that of an isolated cylinder in a steady cross-flow. Nevertheless it is worth noting that, for example, Mureithi et al. (1994a,b) observed a sequence of bifurcations, including period-doubling, flip and pitchfork bifurcations in tube responses, as the flow velocity was increased, leading to chaotic motion. Some of these phenomena may occur also when the VIV of an isolated cylinder is in some way constrained. This paper presents measurements of the VIV of a cylinder in three configurations. After a description of the experimental setup in Section 2, it is shown that the results of tests with a purely linear elastic support system are in close agreement with published data. The response of the cylinder in the presence of end-stops placed first on one side, and then symmetrically on both sides, is discussed in Sections 3 and 4. Depending on the offset and the reduced velocity, the cylinder may exhibit periodic or chaotic motion, with corresponding changes in the magnitude and frequency of impacts. Large displacements in the opposite direction may occur when there is only one end-stop present. Section 5 is concerned with the calculation of lift coefficients from the measurements, and the problem of predicting the motion if the forcing were known. 2. Experimental details 2.1. Description of the apparatus To obtain low turbulence conditions, these experiments were carried out in a towing tank, which was 17 m long, 0.42 m wide, with a water depth of 0.7 m. The design of the remainder of the apparatus was aimed at achieving a low mass ratio and low structural damping. The cylinder, a sealed acrylic tube, had an outer diameter D of 30 mm, a wall thickness of 3 mm and a length of 300 mm between 2.5D diameter endplates. It was mounted horizontally on the elastic system sketched in Fig. 1, and when stationary its axis was 5.5D beneath the free surface. As can be seen, the cylinder was attached beneath a vertical frame which was connected to one end of each of three parallel beams (two towards the bottom and one at the top), mounted at the other end on the carriage. The beams (aluminium strips 1.5 mm 15 mm 920 mm—from which the vertical frame supporting the cylinder was also constructed) remained above the water surface at all times. The system was balanced, making the beams horizontal when the rig was stationary, by adjusting tension springs that connected the top of the vertical frame to the carriage. A photograph of the rig on the carriage is shown in Fig. 2. The frequency of the first mode of oscillation of the system (shown in Fig. 1), with the cylinder in still water, was measured at 0.977 Hz. Subsequent modes and modal frequencies were computed by means of finite elements. The next two, at 2.7 Hz and 8.1 Hz, are torsional modes involving axial oscillations of the cylinder, that would be heavily damped by its endplates. The fourth and fifth modes (8.4 Hz and 9.4 Hz) involve primarily flexing of the beams within their length, and remaining modal frequencies are all above 10 Hz. There was no evidence of any rocking motion in any of the tests. For a vertical displacement of one diameter (30 mm) the horizontal displacement of the cylinder in the first mode of oscillation was less than 0.6 mm. The calculated fundamental natural frequency of the submerged cylinder between pinends was 150 Hz. In the experiments described here, the mass ratio mn (including the effect of contributions from all moving parts) was 1.74. With carriage speeds U between 0.1 m/s and 0.45 m/s the corresponding range of reduced velocities Ur ¼U/fnD was from about 3 to 15. The structural damping in air was 0.2% of critical damping. In water, additional system damping was generated by viscous shear stresses acting on the cylinder’s end plates and on immersed parts of the vertical beams supporting it. Measuring this contribution would entail replacing the cylinder by a horizontal tie bar, adding ballast to represent its inertia and adjusting the balancing springs to return the support system to the same zero
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Fig. 1. Sketch of the cylinder support system showing a side view on the left, and a section of the tank on the right. The mounting points for the beams, the balancing springs and the position sensor were fixed on the carriage, which towed the whole apparatus along the tank at constant speed. Dashed lines show the deflected shapes of the beams in the first mode of oscillation for an amplitude of one diameter.
Fig. 2. Apparatus on the carriage.
position—alterations that would themselves inevitably have an effect on the damping. An alternative approach, found in earlier experiments to provide a good estimate of low levels of hydrodynamic damping (Chaplin, 2000), was to use Stokes’s pffiffiffiffiffiffiffiffiffiffiffiffi formula t ¼ ru on=2 for the shear stress on a plate oscillating with velocity u in its own plane in an unbounded fluid. At a representative frequency of 1 Hz, this led to an estimate of the hydrodynamic damping on all submerged components (except the cylinder itself) of 0.59% of critical damping, suggesting that the effective structural damping z was about 0.79%, or mnz ¼0.014. The system stiffness k1 in this unmodified configuration was measured at 65 N/m per metre length of cylinder. The vertical displacement of the cylinder was recorded by means of a Hamamatsu C2399 optical position sensor system consisting of an active infrared LED mounted on the oscillating frame, and a camera on the carriage. 2.2. Validation of the response in the case of linear structural restraints Before installing end-stops to limit the motion, measurements of VIV were checked against published results. Representative time series of the cylinder displacement y are plotted in Fig. 3, with the corresponding spectra alongside. Peak amplitudes and frequencies are shown in Fig. 4, with data from Khalak and Williamson (1997) for mn ¼2.4 (somewhat higher than the present value of 1.74), and mnz ¼0.014 (as in the present case). In Fig. 4 Anmax is the maximum amplitude of response divided by the cylinder diameter D, and values of fn denote dominant spectral frequencies divided by its natural
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Fig. 3. On the left, displacement time series for vibrations without end-stops, on the right, corresponding spectra.
frequency in still water fn. The results are generally in good agreement with expectations, displaying the initial, upper and lower branches that are characteristic features of the VIV response in these conditions. In the first part of the initial branch, two peaks appear in the spectrum (Fig. 3(a)), at fn (the cylinder natural frequency in still water) and at a frequency denoted fvs. At slightly higher reduced velocities (Fig. 3(b)), the motion is nearly periodic, and there is only one significant peak in the spectrum, at fvs, with Anmax o0:4. The upper branch starts at Ur ¼4.78 when the frequency ratio fnof the periodic motion shown in Fig. 3(c) jumps to unity, and the maximum amplitude jumps to about 0.8D. The motion remains periodic in the first part of the upper branch and the dominant frequency increases slightly with Ur, a characteristic feature of VIV with low mass ratios and low damping (Khalak and Williamson, 1997). n At Ur ¼7.16, a second peak at a frequency denoted f lower appears in the frequency distribution, revealing the start of the transition between upper and lower branches. Over this transition, the cylinder switches intermittently between the two modes, as shown by the motion in Fig. 3(d) for Ur ¼7.68. With increasing Ur, the proportion of time spent by the cylinder in the upper branch mode of vibration diminishes until only the lower mode of oscillation is evident, at Ur ¼8.88 (Fig. 3(e)). In the subsequent lower branch the motion is periodic, with an amplitude and frequency remarkably constant, with n An E0.55 and fn ¼ 1.48, close to the value of 1.51 predicted by the formula for f lower proposed by Khalak and Williamson n (1999) for m ¼1.74 and an added mass coefficient Ca ¼ 1. The reduced velocity where desynchronisation occurs is difficult to identify, but a peak at fvs appears in the frequency distributions at Ur ¼11.44. For higher velocities, the motion becomes much more complex and disorganised, leading to the presence of several peaks in the much wider frequency spectra, including those at fn, fvs and flower. Instabilities can sometimes give rise to a few high amplitude cycles, like those seen in Fig. 3(f) for Ur ¼ 12.63 at tE10 s. The measured values of fvs correspond to a Strouhal number St ¼0.18 which is rather lower than Khalak and Williamson’s (1997) result of 0.22, but close to that in other experiments, e.g. those for a smooth cylinder by Hover et al. (2001) at mnz ¼0.096. Overall, the measurements of VIV obtained with this linear experimental setup show good agreement with comparable results in the literature. It exhibited exactly the characteristic low-mass ratio and low-damping amplitude and frequency responses expected in such cases.
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Fig. 4. (a) Non-dimensional peak frequencies fn and (b) maximum amplitudes Anmax , plotted as functions of reduced velocity for vibrations without endstops: ’, present results with mn ¼ 1.74, mnz ¼ 0.014; J, data from Khalak and Williamson (1997) with mn ¼2.4, mnz ¼ 0.014. The solid line in (a) represents a Strouhal number of 0.22, the broken line, 0.18.
3. VIV with impacts on a single end-stop 3.1. Experimental arrangements Following the validation tests, an end-stop was added to the system to limit the motion on one side, at an offset y/D ¼ eo 0. The end-stop, a short steel bar mounted on the carriage, did not limit the motion of the cylinder directly, but restricted the displacement of its supporting frame above, as can be seen in Fig. 1 (where two symmetrical end-stops are shown). Owing to flexibility in the frame, the effect of contacting an end-stop was therefore that of a sudden increase in structural stiffness rather than a rigid limit. This increased stiffness k1 þk2 was measured at approximately 22.3 kN/m per metre length of cylinder. This is illustrated in Fig. 5, where the nonlinear cases studied by Hover and Triantafyllou (1998) and Stappenbelt and Thiagarajan (2004) are shown as broken lines. In the present experiments three series of tests were carried out, with the end-stop placed in turn at e¼ 0.65 (shown), 0.46 and 0.22.
3.2. Modification of the cylinder dynamics Representative time series of displacements are shown in Fig. 6(a)–(d) for Ur ¼5.46. For the linear elastic case (Fig. 6(a)) this is within the upper branch where the cylinder exhibits a periodic oscillation with amplitude of about 0.8D. When an end-stop is placed at 0.65D the cylinder impacts on it at every cycle, but the result is a stable motion with little amplitude modulation. The same applies at e¼ 0.46, and in both these cases the peak excursion of the cylinder on the side away from the end-stop is reduced to a displacement that almost mirrors the position of the end-stop. At e¼ 0.22 however there is a dramatic change, and the response becomes much more complicated. Through this sequence there is a slight reduction in the dominant frequency from fn ¼1.08 for the unrestricted case, to fn ¼0.99 (St ¼0.18) for e¼ 0.22. At a reduced velocity of 10.9 (Fig. 6(e) and (f)), the unrestricted motion is in the lower branch, and has an almost constant amplitude of about 0.5D. The effect of an end-stop at e¼ 0.22, is to produce a motion with strongly modulated amplitudes, and peak excursions that extend much further, to 1.6D, on the side away from the end-stop. There is also a marked increase in the dominant frequency from fn ¼1.51 (in the unrestricted case) to fn ¼1.85 (St¼ 0.17). Bifurcations diagrams (Fig. 7) reveal the ranges of reduced velocities over which impacts occur, and confirm as expected that virtually all changes in direction from negative to positive velocities take place in the region yo0. They show the stability of 1-impact-per-period motions over most of the impacting domain in all cases. Windows of chaotic 1-impact-per k cycles motions, in which the cylinder describes k cycles of oscillation between one impact and the next, with k varying widely, occur at the start of each impacting domain. For e¼ 0.65, this window is very narrow. It is wider for e¼ 0.46,
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Fig. 5. The elastic resistance of the system, with (a) one end-stop and (b) two end-stops shown as continuous lines. The system characteristics in experiments by Hover and Triantafyllou (1998) and Stappenbelt and Thiagarajan (2004) are shown as broken lines, scaled to match the inner stiffness in the present experiments.
Fig. 6. On the left, time series of dimensionless displacements recorded at Ur ¼ 5.46, (a)–(d), and at Ur ¼10.9, (e)and (f), with the positions of the single end-stop (when present) as indicated. Corresponding spectra are plotted on the right.
and wider still for e¼ 0.22, extending up to Ur ¼7. Windows of chaotic impacting motions can be observed for e¼ 0.46 for Ur 410 and for e¼–0.22 for Ur 412. Fig. 7(c) includes measurements of VIV amplitudes by Stappenbelt and Thiagarajan (2004) with linear and nonlinear elastic supports. Their mass-damping ratio was considerably higher than in the present experiments, and even with the linear system, displacements did not exceed An ¼0.45. Nevertheless, in both experiments, amplitudes observed with nonlinear restraints reached a maximum at reduced velocities higher than in the linear case.
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Fig. 7. Bifurcation diagrams of peak excursions in both directions for single-sided impact cases: (a) e¼ 0.65; (b) e¼ 0.46; (c) e ¼ 0.22. In each plot, results for the linear elastic system are shown as grey symbols. Plot (c) includes measurements of response amplitudes by Stappenbelt and Thiagarajan (2004) with mnz ¼0.222 (considerably higher than in the present experiments): linear system and &, their most nonlinear system (see Fig. 5).
Fig. 8. Peak frequencies of single-sided impacting motions. The line represents a Strouhal number of 0.18.
Peak frequencies in the response for impacting motions at all offsets are plotted in Fig. 8. Many points lie close to the line representing a Strouhal number of 0.18, though in these conditions for the smallest offset there was a wider bandwidth, and in some cases two or even three distinct dominant frequencies. An example appears in Fig. 6(f), reflecting the beating motion that can be seen in the time series. The perturbation of the dynamics of the cylinder caused by the presence of an end-stop can also be observed by means of phase portraits, plotting the trajectory of the system in the plane _ c Þ, as shown in Fig. 9. Many of these features described above can be seen here also. ðy=D, y=U 4. VIV with impacts on two symmetrically-placed end-stops Four series of tests were conducted with end-stops at e¼ 70.20, 70.30, 70.47 and 70.65. For e¼ 70.65, impacts occurred only over the range of reduced velocities corresponding to the upper branch in the linear case. For e¼ 70.47, it occurred over the upper and lower branches, and for e¼ 70.30 and 70.20 it occurred over nearly all the lock-in domain. Representative time series of dimensionless displacements y/D recorded at Ur ¼5.12 are plotted in Fig. 10. For e¼ 70.65 and 70.47, the presence of the end-stops does not seem to affect the periodic nature of the motion, and with very few exceptions the cylinder impacts once on each stop during each cycle of oscillation.
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_ c axes have the same Fig. 9. Phase portraits for the linear elastic case and for single-sided impacting motions. All y/D axes have the same scale, and all y=U scale. The motion is clockwise around each trajectory.
When the end-stops are moved in, in this case to e¼ 70.30 or 70.20, the cylinder displacement exhibits sequences of symmetric double-impact motions, but also cases where it impacts successively on the same stop or where it undergoes an oscillation near the mid-point. The motion in these cases is much more affected by the end-stops and the response is more complex without any distinctive periodic pattern. The frequency content of the cylinder displacement also shows the perturbation of the dynamics caused by the presence of the end-stops. As illustrated by the frequency ratio distributions presented in Fig. 11, the magnitude of the main peak diminishes as e decreases, while those in the side bands increase. In Fig. 11, frequency distributions for the linear elastic case are presented in the first column, while those in the other columns correspond to double-sided impacting motions with the offset decreasing while going to the right. The reduced velocity increases when going down a row. The energy becomes more widely distributed as the motion is more constricted. Frequency distributions for motions with small end-stop offsets, exhibiting some structure but with energy distributed over wide ranges of frequencies, provide an indication of chaotic motions. Even though the motion appears to be substantially unaffected by impacts on end-stops at e¼ 70.65 in Fig. 10, for most reduced velocities its frequency was not the same as in the linear elastic case. In Fig. 12 the peak frequency is plotted as a function of Ur and in fact is seen to rise at a rate corresponding to a constant Strouhal number, without lock-in, for all double-sided impacting cases over their respective impacting domains. The absence of lock-in is consistent with the results of Hover and Triantafyllou (1998), in which (with weaker nonlinearities—see Fig. 5), the lock-in range was considerably narrower than in the linear case. Characteristic motions seen in the phase portraits in Fig. 13 can be classified using the identification k-‘-m where in k cycles the cylinder impacts ‘ times on the upper stop and m times on the lower stop. For e¼ 70.65 at the reduced velocities shown, the cylinder exhibits a stable 1-1-1 motion. When the offset is reduced below e¼ 70.65, the trajectories become less repeatable. This trend continues as e is reduced, and at e¼ 70.30 and 70.20 there is barely any discernable pattern and the trajectories fill most of the phase plane enclosed by their widest and narrowest cycles, again indicative of a motion of a chaotic nature. Nevertheless some characteristic features can be identified. At low reduced velocities the cylinder amplitude is strongly decreased after an impact and the cylinder may describe several cycles of amplitude growth before its next impact. An example of such a sequence, taken from the test with Ur ¼4.44 and e¼ 70.20, is shown in Fig. 14(a), where the cylinder can be observed describing a k-0-1 motion with k ¼4, i.e. four oscillation cycles after leaving the lower stop before reaching the lower stop again. Under these conditions k varied between 2 and 10 without any apparent pattern. The maximum period of these motions, largest after the first impacts, decreases as Ur increases. In Fig. 14(b) (Ur ¼4.78) the motion is 2-0-1, while in Fig. 14(c) (Ur ¼7.51) the cylinder follows successive anti-symmetric 1-1-1 cycles, the second one shown as a dashed line. Also, trajectories with successive bounces on the same end-stop occurred for cases with small offsets and high reduced velocities. Examples are shown in Fig. 14(d) (Ur ¼10.92) and in Fig. 15 (Ur ¼11.95), where it can be seen that the cylinder sometimes chattered against the end-stops for as long as one complete period of end-to-end oscillation. In those cases in which a distinct frequency can be detected in the chattering motion, it is above 30 Hz.
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Fig. 10. Displacement time series at Ur ¼ 5.12: (a) for the linear elastic case, and for double-sided impacting motions at (b) e ¼ 70.65; (c) e¼ 7 0.47; (d) e¼ 70.30; (e) e¼ 7 0.20.
Fig. 11. Frequency distributions for the linear elastic case (on the left) and for double-sided impacting motions.
Bifurcation diagrams of lower peak excursions, showing points on the (Ur,y/D) plane at which, in individual cycles, the trajectory crossed the Poincare´ section ðy, y_ ¼ 0, tÞ from y_ o 0 to y_ 4 0 are shown in Fig. 16. In these diagrams the points show impacts on the lower stop as well as positions between the stops at which the cylinder reversed direction from down
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Fig. 12. Peak frequencies in double-sided impacting motions. The line represents St ¼0.18.
Fig. 13. Phase portraits for linear restraints and double-sided impacting motions. Broken lines represent the positions of the end-stops. The motion is clockwise around each trajectory.
Fig. 14. Phase portraits taken from the motions at e ¼ 7 0.20 and reduced velocities Ur of (a) 4.44, (b) 4.78, (c) 7.51 and (d) 10.92.
to up. For e¼ 7 0.65 the bifurcation diagram shows the upper branch being transformed into a window of stability of the 1-1-1 motion, since reversals take place only at the stop position. The 1-1-1 motion is also dominant for e¼ 70.47 over the entire impacting domain, even if some lower peak excursions scattered over the negative part of the bifurcation diagram reveal some loss of stability. When e is decreased to 70.30, numerous lower peak excursions scattered over the negative and positive sides of the bifurcation diagram reveal the complex nature of the motions in this case. The increased density of points close to the upper stop position at the end of the impacting domain, for Ur 410, shows the occurrence of successive impacts on the upper stop, or 1-‘-0 trajectories with ‘ 4 1. For the lowest offset value, lower peak excursions are densely spread across the bifurcation diagram, revealing the complex nature of the motion over the impacting domain in this case. Successive impacts on the upper stop or chattering
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Fig. 15. Displacement time series for e¼ 7 0.20, Ur ¼11.95.
Fig. 16. Bifurcation diagrams of lower peak excursions for double-sided impacting motions. (a) e ¼ 7 0.65, (b) e¼ 70.47, (c) e¼ 7 0.30 and (d) e¼ 70.20. Symbols in grey refer to the case without end-stops.
motions start to occur in this case much earlier than at e¼ 70.30, at around Ur ¼8. It is interesting to note that, once started, impacts continue to occur on end-stops placed beyond the peak excursion of the cylinder in the case without endstops (e.g. in Fig. 16(b)).
5. Lift coefficients 5.1. A single end-stop When the cylinder encounters or leaves an end-stop it experiences a sudden large change in structural stiffness. The subsequent motion (until the cylinder next leaves or meets an end-stop) is that of a forced single degree of freedom system, released from a defined starting position with a defined velocity. In such a case, if the forcing F(t)were known (where t is measured from the instant at which the change in stiffness takes place), the motion y(t) up to the time of the next impact could be found by means of the Laplace transform of its equation of motion pffiffiffiffiffiffiffi my€ þ2z mky_ þky ¼ FðtÞ, ð1Þ
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using the known initial displacement and velocity as the two required boundary conditions. For present purposes it seems reasonable to represent the fluid forces in the form F ¼ 12rU 2 d½C Y0 þC Y cosðotaÞ,
ð2Þ
in which a time-independent component (represented by CY0) might appear because of the asymmetric nature of the motion when only one end-stop is present. The solution of (1) subject to forcing of the form of (2) with y¼y0 and y_ ¼ y_ 0 at time t ¼0 is given in the appendix. Apart from the time independent term it consists of apcomponent of constant amplitude ffiffiffiffiffiffiffiffiffiffi at the driving frequency o, and a decaying component at the natural frequency o0 ¼ k=m. If the structural damping is light, it has little effect on the initial motion (say over a time interval of 2p/o), which can then be approximated very closely by putting z ¼0 in Eq. (A2). This is then simply the sum of two sinusoids and a constant term. Matching this formula to the observed displacement of the cylinder between end-stop contacts provides a way of estimating lift coefficients for the impacting motions described above. In the case of a cylinder undergoing VIV on elastic supports without restraints, lift coefficients can be computed from displacement records by following the approach used by Khalak and Williamson (1999). This consists basically of reorganising Eq. (1) (with m representing the effective mass of the cylinder) and (2) (without CY0), to find those values of CY cos a and CY sin a such that both the peak displacement, and the frequency indicated by the sum of the system’s inertia and the force in anti-phase with the acceleration (represented by CY cos a), match the measurements. In that situation the choice of the meaning of m in Eq. (1) is largely a matter of convention, since changing to any other definition would be accommodated by an adjustment in the value of CY cos a. In the present case, when trying to match the measured response to Eqs. (2), this freedom does not exist, because the response is likely to contain a distinct component at frequency p(1) ffiffiffiffiffiffiffiffiffiand ffi o0 ¼ k=m. Therefore m in Eq. (1) should include the effective added mass, which is initially unknown. To analyse the present data, a least squares procedure was applied over each interval between the cylinder leaving the end-stop and its next impact, with the aim of minimising the root-mean-square difference E between the measured displacements over the duration of a complete run, and the motion obtained from Eq. (A2) on a cycle-by-cycle basis. For a given value of the added mass coefficient Ca (making m¼md(mn þCa) where md is the displaced mass) the variables to be evaluated in the least squares analysis are the forces F 0 ¼ 12rU 2 C Y0 , F C ¼ 12rU 2 C Y cos a and F S ¼ 12rU 2 C Y sin a. The driving frequency o in each case was taken to be that which corresponds to a Strouhal number of 0.18, matching the trends seen in Figs. 8 and 12. It might be expected that a least squares approach could deliver also Ca for each run, but the system of equations is not sufficiently well conditioned to provide robust solutions with four unknowns. Instead, a constant value was adopted throughout. This was based on results like those in Fig. 17, in which the root-mean-square error (with best-fit values for the other variables) for each of six cases is plotted as a function of the trial value of Ca. The average position of the minimum error over all cases was Ca ¼0.65, and this was used in subsequent calculations. Displacements, computed from Eq. (A2) with the amplitude of each harmonic component determined by least-squares analysis, are generally in excellent agreement with the measured data, as shown in Fig. 18. In most cases the measurements (plotted as continuous lines) are practically indistinguishable from the least-squares fit (broken lines), except where there are several oscillations between one impact and the next (e.g. in Fig. 18(d) and (e)). It seems reasonable to assume that this is because the vortex-induced forces are strongly modified when the cylinder is no longer undergoing sudden reversals at frequent intervals. Mean lift coefficients computed from the least-squares fit are plotted in Fig. 19. A marked difference from the case of unrestrained VIV (Fig. 19(a)) is the absence of high values of CY at Ur E4.7. This is normally the beginning of the upper branch of the response, but as noted above that does not appear when end-stops are limiting the motion. On the other hand, lift coefficients for impacting motions tend to rise at higher reduced velocities, unlike those in the case of VIV with linear restraints. The mean force, normalised as CY0, was extremely small in all cases. In the case of VIV with linear restraints, if one were to integrate the cylinder’s equation of motion (1) using experimentally obtained force coefficients for the forcing on the right hand side, it would be reasonable to expect that the
Fig. 17. Root-mean-square errors in predictions of displacements obtained with Eq. (A2).
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Fig. 18. Displacement time series: continuous lines are measurements, broken lines from Eq. (A2) with optimal coefficients and Ca ¼ 0.65. Each impact on the end-stop is identified with a symbol. (a) e¼ 0.65, Ur¼5, (b) e¼ 0.65, Ur¼8.0, (c) e¼ 0.46, Ur¼5.1, (d) e¼ 0.46, Ur¼11.1, (e) e¼ 0.22, Ur¼5.5 and (f) e¼ 0.22, Ur¼11.9.
computed motion would closely match the measurements, at least in those conditions where the motion is strongly periodic. Moreover, in such cases, where the force coefficients are smooth functions of the reduced velocity, the same would apply to computed amplitudes. This is quite different from the behaviour of an impact or bi-linear oscillator responding to harmonic forcing. Much of the literature on that subject is concerned with the consequences of the extreme sensitivity of such systems to the controlling parameters, and the presence of bifurcations in their response (e.g. Foale and Bishop, 1994; Natsiavas, 1990). Response amplitudes obtained by numerically integrating the equation of motion of a system that incorporates sudden changes in stiffness like those in Fig. 5 do not in general exhibit the smooth qualities of the linear case. Small changes in the amplitude or frequency of the forcing, or in the starting conditions, may cause the motion to jump from one type of response to another. To check how well the displacements of the cylinder measured in the present experiments could be recovered from a knowledge of the characteristics of the elastic system and the best-fit lift coefficients, a Newmark-b scheme (with b ¼1/6 and g ¼1/2) was used to integrate the cylinder’s equation of motion, with a bisection routine to identify the times of contact with, and separation from, the end-stop, and an appropriate change in stiffness at each transition. Test runs were carried out to validate the code, by comparing results with those obtained from the analytical solution for the same problem developed by Bapat (2008, 2011). Typical computations advanced in as many as 10 000 time steps in every cycle
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Fig. 19. Mean lift coefficients for the case of no end-stops, and for single-sided impacting motions. In (a), data from Khalak and Williamson (1997) are shown with unfilled symbols. (a) No end-stops, (b) e¼ 0.65, (c) e¼ 0.46 and (d) e¼ 0.22.
Fig. 20. Average impact velocities on a single end-stop: þ , measurements; , results of a numerical time-stepping model of the motion of the cylinder, computed using lift coefficients provided by the polynomials plotted in Fig. 19 with dashed lines. (a) e¼ 0.65, (b) e¼ 0.46 and (c) e¼ 0.22.
of the forcing, and reached a steady state before 1000 cycles had been completed. The process was started with the system at rest at zero offset, and the forcing ramped up over 20 cycles. Numerical results agreed with those obtained from Bapat’s analytical approach to at least 5 decimal places. The same programme was used to model the response of the cylinder at each of a large number of reduced velocities. In each case the frequency of the forcing corresponded to a Strouhal number of 0.18, and its amplitude was derived from a polynomial (shown as a dashed line for each end-stop position in Fig. 19) fitted to the measured lift coefficients. Some
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Fig. 21. Average impact velocities on two symmetrical end-stops: þ , measurements; , results of the numerical time-stepping impact oscillator, using the same lift coefficients as in Fig. 20, i.e. those based on the measurements of single-sided impacts. (a) e¼ 0.65, (b) e¼ 0.47 and (c) e¼ 0.20.
results are plotted in Fig. 20, in the form of the mean velocity of the cylinder at the instant of impact on the end-stop, normalised with respect to Uc ¼2pfn/D. Numerical results are shown as solid grey circles. As expected, over some ranges of reduced velocity, there are bifurcations in the computed response. There are also regions, Ur 46.8 at e¼ 0.65, Ur 47.9 at e¼ 0.46 and 9.7oUr o11.4 at e¼ 0.22, where no impacts occur. When seen in this light, jumps from one branch to another can be detected also in the measured impact velocities (plotted in Fig. 20), at about Ur ¼ 5.5 for all three end-stop positions. In most cases the velocities are in reasonable agreement with the numerical results, though impacts occurred in conditions where in the numerical model there was no contact. Differences between measured and predicted velocities may be associated with the absence, in the numerical modelling, of any feedback from the cylinder motion to the forcing. It seems reasonable to assume that the very high acceleration of the cylinder that occurs during every impact should lead to a departure, at least momentarily, from a purely harmonic lift force. It would be necessary to model the flow in order to include this effect in numerical predictions. 5.2. Symmetrical end-stops Over the short interval between impacts on symmetrically placed end-stops, there was not enough information from the cylinder trajectory to support the calculation of force coefficients as attempted above. Because the motion was more restricted, impact velocities were lower than in the case of a single end-stop, and are plotted in Fig. 21. They are compared there with predictions of the numerical impact oscillator model, using the same lift coefficients as before. Again there is some measure of agreement, but flow modelling would be required to achieve a better match. 6. Conclusions Measurements have been carried out of the vortex-induced cross-flow vibrations of an isolated stiff cylinder mounted on an elastic system in the presence of end-stops that introduced a very large increase in the stiffness of the system at a particular offset, either on one side or on both sides of the cylinder’s initial position. The mass ratio was 1.74 and structural damping was 0.79% of critical. In the absence of the end-stops, measurements of VIV were shown to be in close agreement with previous data for systems with similarly low mass ratio and low damping. Placing end-stops at an offset of 0.65 diameters, on one or both sides, eliminated any evidence of frequency lock-in. The dominant frequency in the response remained close to that indicated by a constant Strouhal number of 0.18 over the range of reduced velocities between about 4 and 14, and the motion was characterised by a single impact on each end-stop
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in every period. More complicated displacement time histories appeared with offsets at 0.3 diameters or less, when the cylinder exhibited successive impacts on the same end-stop, reversals near its mean position, and chattering on one of the end-stops for as long as one complete period of oscillation. Response spectra reveal a progressive growth in sub- and superharmonics, as well as a general broadening of the frequency content. When there were regular impacts on a single end-stop, the motion of the cylinder between impacts was shown to match the expected trajectory of a forced linear single-degree-of-freedom system released from an initial offset with a given velocity. It was then possible to extract lift coefficients and compare measured impact velocities with those computed from a numerical model of an impact oscillator, using the same forcing. The results were rather mixed, but illuminated the existence of jumps in the measurements between modes of response. These and other features of the observed response are indicative of a degree of chaos that increases as the motion of the cylinder is more and more confined, in conditions that give rise to lock-in in the linear elastic case. This was confirmed by computations of the Lyapunov exponent that are beyond the scope of the present paper but are reported in Bourdier (2008). At reduced velocities around 10, the motion of the cylinder when constrained on both sides appeared more chaotic than when a single end-stop was present, as can be seen in bifurcation diagrams. In the latter case the cylinder impacted on the end-stop once in every cycle, but rebounded with an excursion of as much as one and a half diameters beyond its stationary position in the opposite direction. In both cases mean impact velocities increased with the reduced velocity, and maximum mean values were close to Uc ¼2pfn/D, or in other words comparable to the maximum velocity of a cylinder undergoing lock-in VIV with linear elastic compliance. Further work could usefully be carried out on the effect of other forms of nonlinearities in the support system, and, in the present configuration, on the influence of the stiffness of the end-stops. A companion paper (Bourdier and Chaplin, in preparation) compares the present measurements with numerical predictions based on wake oscillator models. Appendix A Consider a single-degree-of-freedom system pffiffiffiffiffiffiffi my€ þ 2z mky_ þ ky ¼ F 0 þ F c cos ot þF s sin ot,
ðA1Þ
released at time t ¼0, from a position y¼ y0 with a velocity y_ ¼ y_ 0 . Then by means of a Laplace transform of Eq. (A1) it can be shown that the ensuing motion is y¼
F 0 ð1O2 ÞF c 2zOF s 2F c Oz þ ð1O2 ÞF s SF 0 þ ð1O2 ÞF c 2zOF s kSy0 zo0 t 0 þ cos ot þ sin ot e cos z o0 t k kS kS kS
zSF 0 þ zð1O2 ÞF c þ Oð12z2 O2 ÞF s Sðkzy0 þmo0 y_ 0 Þ zo0 t 0 e sin z o0 t, z0 kS
ðA2Þ
qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 0 2 where o0 ¼ k=m, O ¼ o/o0, z ¼ 1z and 2
S ¼ 12O2 þ4O2 z þ O4 :
ðA3Þ
It can be seen that the response consists of a steady offset, a term of constant amplitude at the driving frequency o and a term of decaying amplitude at frequency o0. References Bapat, C.N., 2008. Exact solution of stable periodic one contact per N cycles motion of a damped linear oscillator contacting a unilateral elastic stop. Journal of Sound and Vibration 314, 803–820. Bapat, C.N., 2011. Correction of Misprints in Bapat (2008). Personal Communication. Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics 16, 195–222. Bearman, P.W., 2011. Circular cylinder wakes and vortex-induced vibrations. Journal of Fluids and Structures 27, 648–658. Bourdier, S., 2008. Vortex-Induced Vibrations of a Nonlinearly Supported Rigid Cylinder. Ph.D. Thesis. University of Southampton, UK. Bourdier, S., Chaplin, J.R. Vortex-induced vibrations of a rigid cylinder on elastic supports with end-stops. Part 2: numerical modelling and comparison with measurements, in preparation. Chaplin, J.R., 2000. Hydrodynamic damping of a cylinder at b E106. Journal of Fluids and Structures 14, 1101–1117. Foale, S., Bishop., S.R., 1994. Bifurcations in impact oscillations. Nonlinear Dynamics 6 (3), 285–299. Gelbe, H., Jahr, M., Schroder, K., 1995. Flow-induced vibrations in heat-exchanger tube bundles. Chemical Engineering and Processing 34 (3), 289–298. Hover, F.S., Triantafyllou, M.S., 1998. Lock-in phenomena for cylinders with nonlinear compliance. In: Proceedings of FEDSM98, ASME Fluids Engineering Division Summer Meeting. Washington, DC, USA. Hover, F.S., Tvedt, H., Triantafyllou, M.S., 2001. Vortex-induced vibrations of a cylinder with tripping wires. Journal of Fluid Mechanics 448, 175–195. Ibrahim, R.A., 2011. Mechanics of pipes conveying fluids—part II: applications and fluidelastic problems. ASME Journal of Pressure Vessel Technology 133 (2) (article number 024001). Kevlahan, N.K.-R., 2011. The role of vortex wake dynamics in the flow-induced vibration of tube arrays. Journal of Fluids and Structures 27, 829–837. Khalak, A., Williamson, C.H.K., 1997. Fluid forces and dynamics of a hydroelastic structure with very low mass and damping. Journal of Fluids and Structures 11, 973–982. Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping parameter. Journal of Fluids and Structures 13, 813–851. Mureithi, N.W., Price, S.J., Paı¨doussis, M.P., 1994a. The post-Hopf-bifurcation response of a loosely supported cylinder in an array subjected to crossflow—part 1: experimental results. Journal of Fluids and Structures 8, 833–852.
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